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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 4. Statistics & Probability / 4.3 Probability / 4.3.3 Sample Space Diagrams


4.3.3 Sample Space Diagrams


Venn Diagrams

What is a Venn diagram?

  • A Venn diagram is a way to illustrate events from an experiment and are particularly useful when there is an overlap between possible outcomes
  • A Venn diagram consists of
    • a rectangle representing the sample space (U)
      • The rectangle is labelled U 
      • Some mathematicians instead use S or ξ 
    • a circle for each event
      • Circles may or may not overlap depending on which outcomes are shared between events
  • The numbers in the circles represent either the frequency of that event or the probability of that event
    • If the frequencies are used then they should add up to the total frequency
    • If the probabilities are used then they should add up to 1

What do the different regions mean on a Venn diagram? 

  • A apostrophe is represented by the regions that are not in the A circle
  • A intersection B is represented by the region where the A and B circles overlap
  • A union B is represented by the regions that are in A or B or both
  • Venn diagrams show ‘AND’ and ‘OR’ statements easily
  • Venn diagrams also instantly show mutually exclusive events as these circles will not overlap
  • Independent events can not be instantly seen
    • You need to use probabilities to deduce if two events are independent

3-2-1-fig1-venn-and-set-notation

3-1-2-fig2-various-venns-part-2

How do I solve probability problems involving Venn diagrams?

  • Draw, or add to a given Venn diagram, filling in as many values as possible from the information provided in the question
  • It is usually helpful to work from the centre outwards
    • Fill in intersections (overlaps) first
  • If two events are independent you can use the formula
    • straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis straight P left parenthesis B right parenthesis
  • To find the conditional probability straight P left parenthesis A vertical line B right parenthesis
    • Add together the frequencies/probabilities in the B circle
      • This is your denominator
    • Out of those frequencies/probabilities add together the ones that are also in the A circle
      • This is your numerator
    • Evaluate the fraction

plx9zFR~_3-2-2-fig1-set-notation-examples

Exam Tip

  • If you struggle to fill in a Venn diagram in an exam:
    • Label the missing parts using algebra
    • Form equations using known facts such as:
      • the sum of the probabilities should be 1
      • P(A∩B)=P(A)P(B) if A and B are independent events

Worked Example

40 people are asked if they have sugar and/or milk in their coffee. 21 people have sugar, 25 people have milk and 7 people have neither.

a)
Draw a Venn diagram to represent the information.

4-3-3-ib-ai-aa-sl-venn-diagram-a-we-solution

b)
One of the 40 people are randomly selected, find the probability that they have sugar but not milk with their coffee.

4-3-3-ib-ai-aa-sl-venn-diagram-b-we-solution

c)
Given that a person who has sugar is selected at random, find the probability that they have milk with their coffee.

4-3-3-ib-ai-aa-sl-venn-diagram-c-we-solution

Tree Diagrams

What is a tree diagram?

  • A tree diagram is another way to show the outcomes of combined events
    • They are very useful for intersections of events
  • The events on the branches must be mutually exclusive
    • Usually they are an event and its complement
  • The probabilities on the second sets of branches can depend on the outcome of the first event
    • These are conditional probabilities
  • When selecting the items from a bag:
    • The second set of branches will be the same as the first if the items are replaced
    • The second set of branches will be the different to the first if the items are not replaced

How are probabilities calculated using a tree diagram?

  • To find the probability that two events happen together you multiply the corresponding probabilities on their branches
    • It is helpful to find the probability of all combined outcomes once you have drawn the tree
  • To find the probability of an event you can:
    • add together the probabilities of the combined outcomes that are part of that event
      • For example: straight P left parenthesis A union B right parenthesis equals straight P left parenthesis A intersection B right parenthesis plus straight P left parenthesis A intersection B apostrophe right parenthesis plus straight P left parenthesis A apostrophe intersection B right parenthesis
    • subtract the probabilities of the combined outcomes that are not part of that event from 1
      • For example: straight P left parenthesis A union B right parenthesis equals 1 minus straight P left parenthesis A apostrophe intersection B apostrophe right parenthesis

UclzomJM_3-2-3-fig1-tree-setup

Do I have to use a tree diagram?

  • If there are multiple events or trials then a tree diagram can get big
  • You can break down the problem by using the words AND/OR/NOT to help you find probabilities without a tree
  • You can speed up the process by only drawing parts of the tree that you are interested in

Which events do I put on the first branch?

  • If the events A and B are independent then the order does not matter
  • If the events A and B are not independent then the order does matter
    • If you have the probability of A given B then put B on the first set of branches
    • If you have the probability of B given A then put A on the first set of branches

Exam Tip

  • In an exam do not waste time drawing a full tree diagram for scenarios with lots of events unless the question asks you to
    • Only draw the parts that you are interested in

Worked Example

20% of people in a company wear glasses. 40% of people in the company who wear glasses are right-handed. 50% of people in the company who don’t wear glasses are right-handed.

a)
Draw a tree diagram to represent the information.

4-3-3-ib-ai-aa-sl-tree-diagram-a-we-solution

b)
One of the people in the company are randomly selected, find the probability that they are right-handed.

4-3-3-ib-ai-aa-sl-tree-diagram-b-we-solution

c)
Given that a person who is right-handed is selected at random, find the probability that they wear glasses.

4-3-3-ib-ai-aa-sl-tree-diagram-c-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
                                                                    • 3.1.3 Arcs & Sectors
                                                                    • 3.2 Geometry of 3D Shapes
                                                                      • 3.2.1 3D Coordinate Geometry
                                                                        • 3.2.2 Volume & Surface Area
                                                                        • 3.3 Trigonometry
                                                                          • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                            • 3.3.2 Non Right-Angled Trigonometry
                                                                              • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                              • 3.4 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 4. Statistics & Probability
                                                                                                  • 4.1 Statistics Toolkit
                                                                                                    • 4.1.1 Sampling & Data Collection
                                                                                                      • 4.1.2 Statistical Measures
                                                                                                        • 4.1.3 Frequency Tables
                                                                                                          • 4.1.4 Linear Tranformations of Data
                                                                                                            • 4.1.5 Outliers
                                                                                                              • 4.1.6 Univariate Data
                                                                                                                • 4.1.7 Interpreting Data
                                                                                                                • 4.2 Correlation & Regression
                                                                                                                  • 4.2.1 Bivariate Data
                                                                                                                    • 4.2.2 Correlation & Regression
                                                                                                                    • 4.3 Probability
                                                                                                                      • 4.3.1 Probability & Types of Events
                                                                                                                        • 4.3.2 Conditional Probability
                                                                                                                          • 4.3.3 Sample Space Diagrams
                                                                                                                          • 4.4 Probability Distributions
                                                                                                                            • 4.4.1 Discrete Probability Distributions
                                                                                                                              • 4.4.2 Expected Values
                                                                                                                              • 4.5 Binomial Distribution
                                                                                                                                • 4.5.1 The Binomial Distribution
                                                                                                                                  • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                  • 4.6 Normal Distribution
                                                                                                                                    • 4.6.1 The Normal Distribution
                                                                                                                                      • 4.6.2 Calculations with Normal Distribution
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        Daniel Finlay

                                                                                                                                                                        Author: Daniel

                                                                                                                                                                        Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.


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